Solve $x^2 + 14x = -24$ By Completing The Square

To solve the quadratic equation $x^2 + 14x = -24$ by completing the square, we need to transform the left side of the equation into a perfect square trinomial. Here's a step-by-step guide on how to do it:

Step 1: Understand Completing the Square

Completing the square is a technique used to rewrite a quadratic expression in the form of $(x + a)^2 + b$, where $a$ and $b$ are constants. This form is particularly useful because $(x + a)^2$ is a perfect square, making it easier to solve for $x$. The main idea is to add a constant to both sides of the equation to create this perfect square trinomial. The perfect square trinomial is a trinomial that can be factored into the square of a binomial. For example, $x^2 + 2ax + a^2$ is a perfect square trinomial because it can be factored into $(x + a)^2$.

The general form of a quadratic equation is $ax^2 + bx + c = 0$. When $a = 1$, the process of completing the square involves adding and subtracting $(\frac{b}{2})^2$ to the left side of the equation. This ensures that the quadratic expression can be factored into a perfect square. Understanding this concept is crucial for applying the completing the square method effectively. Keep in mind that this method is applicable not only for solving quadratic equations but also for rewriting them in vertex form, which is highly beneficial in graphing parabolas and solving optimization problems. By mastering this technique, you gain a powerful tool for manipulating quadratic expressions and solving related problems in algebra and beyond. Let's proceed with applying this to our specific equation, $x^2 + 14x = -24$, to find the solution set.

Step 2: Rewrite the Equation

We start with the given equation:

$x^2 + 14x = -24$

To complete the square, we need to add a value to both sides of the equation that will make the left side a perfect square trinomial. The general form to achieve this is by adding $(\frac{b}{2})^2$, where $b$ is the coefficient of the $x$ term. In our equation, $b = 14$.

Step 3: Calculate the Value to Add

We calculate $(\frac{b}{2})^2$ as follows:

$(\frac{14}{2})^2 = (7)^2 = 49$

This value, 49, is what we need to add to both sides of the equation to complete the square.

Step 4: Add to Both Sides

Add 49 to both sides of the equation:

$x^2 + 14x + 49 = -24 + 49$

This step is crucial because it maintains the balance of the equation while transforming the left side into a perfect square trinomial. Adding the same value to both sides ensures that the solutions to the equation remain unchanged. The value we calculated, 49, is specifically chosen to make the left side factorable as a perfect square. This allows us to rewrite the equation in a form that is much easier to solve. Understanding this balance is key to mastering algebraic manipulations and solving equations effectively. The next step will involve factoring the left side and simplifying the right side, bringing us closer to the solution. This method highlights the elegance of algebraic techniques in transforming complex expressions into simpler, manageable forms. By carefully applying each step, we can systematically solve quadratic equations and gain a deeper understanding of their properties.

Step 5: Factor the Left Side and Simplify the Right Side

Now, we factor the left side, which is a perfect square trinomial, and simplify the right side:

$(x + 7)^2 = 25$

The left side, $x^2 + 14x + 49$, factors neatly into $(x + 7)^2$ because we specifically chose 49 to make this happen. The right side simplifies from $-24 + 49$ to 25. This step transforms the original quadratic equation into a much simpler form, where we have a squared expression equal to a constant. Recognizing and factoring perfect square trinomials is a fundamental skill in algebra, making the subsequent steps straightforward. This transformation is crucial because it allows us to apply the square root property, which is the next step in solving for $x$. The equation $(x + 7)^2 = 25$ is now in a form where we can easily isolate $x$ by taking the square root of both sides. This methodical approach demonstrates the power of algebraic manipulation in simplifying complex problems into manageable steps. The clarity and precision in each step are essential for accurate problem-solving and a deeper understanding of mathematical concepts.

Step 6: Take the Square Root of Both Sides

Take the square root of both sides:

$\sqrt{(x + 7)^2} = \pm\sqrt{25}$

This gives us:

$x + 7 = \pm 5$

Taking the square root of both sides introduces the $\pm$ symbol, which is critical because both positive and negative square roots of 25 satisfy the equation. This is a key concept in solving quadratic equations, as it highlights that there can be two solutions. The equation $x + 7 = \pm 5$ represents two separate equations: $x + 7 = 5$ and $x + 7 = -5$. Each of these equations will lead to a different solution for $x$. Understanding this step is vital for finding all possible solutions to the original quadratic equation. The inclusion of both positive and negative roots ensures that we account for all values of $x$ that make the equation true. This careful consideration of both possibilities is a hallmark of thorough and accurate mathematical problem-solving.

Step 7: Solve for $x$

Now, we solve for $x$ by subtracting 7 from both sides:

$x = -7 \pm 5$

This gives us two possible solutions:

$x = -7 + 5 = -2$

$x = -7 - 5 = -12$

Step 8: Write the Solution Set

Therefore, the solution set is $\{-12, -2\}$.

So, the correct answer is:

A. $\{-12, -2\}$

In summary, solving the quadratic equation $x^2 + 14x = -24$ by completing the square involves transforming the equation into a perfect square trinomial, which simplifies the process of finding the solutions. The steps include adding the square of half the coefficient of the $x$ term to both sides, factoring the perfect square trinomial, taking the square root of both sides, and solving for $x$. This method not only provides the solutions but also enhances the understanding of quadratic equations and their properties. Mastering completing the square is a valuable skill for solving a variety of quadratic equation problems. By following these steps, we systematically arrive at the solution set $\{-12, -2\}$, demonstrating the effectiveness of this algebraic technique.